What is the exact value of cos(120 degrees)?
Understand the Problem
The question is asking for the exact value of the cosine function at an angle of 120 degrees. To solve this, we will use the properties of trigonometric functions, specifically the unit circle.
Answer
$-\frac{1}{2}$
Answer for screen readers
The exact value of $\cos(120^\circ)$ is $-\frac{1}{2}$.
Steps to Solve
- Convert the angle to radians
To work with the cosine function, it's helpful to convert degrees to radians. The conversion formula is: $$ \text{radians} = \frac{\pi}{180} \times \text{degrees} $$ For 120 degrees, it becomes: $$ \text{radians} = \frac{\pi}{180} \times 120 = \frac{2\pi}{3} $$
- Locate the angle on the unit circle
120 degrees (or $\frac{2\pi}{3}$ radians) is in the second quadrant of the unit circle. Here, the cosine value is negative.
- Determine the reference angle
The reference angle can be found by subtracting 120 degrees from 180 degrees: $$ \text{Reference angle} = 180^\circ - 120^\circ = 60^\circ $$ In radians, this is: $$ \text{Reference angle} = \frac{\pi}{3} $$
- Find the cosine of the reference angle
The cosine of the reference angle $\frac{\pi}{3}$ is: $$ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} $$
- Apply the sign for cosine in the second quadrant
Since cosine is negative in the second quadrant, we have: $$ \cos(120^\circ) = -\cos\left(\frac{\pi}{3}\right) = -\frac{1}{2} $$
The exact value of $\cos(120^\circ)$ is $-\frac{1}{2}$.
More Information
The cosine function gives the x-coordinate of the point on the unit circle corresponding to a given angle. For angles in the second quadrant, the cosine values are negative. The reference angle helps us find the cosine value easily.
Tips
- Confusing the quadrants: Remember, cosine is positive in the first quadrant and negative in the second quadrant. Make sure to pay attention to the angle's location.
- Failing to convert degrees to radians: Ensure to always convert to radians when required, especially when dealing with trigonometric functions that are commonly expressed in radians.
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